Since the successful development of the crystalline thermoplastic polyglycolide (PG) and 10/90 poly(1-lactide-co-glycolide) (PLG) as absorbable suture materials, there have been many successful attempts to prepare a number of new absorbable, crystalline homopolymers as well as segmented and block copolymers by ring-opening or condensation polymerization for use in a variety of biomedical and pharmaceutical applications. Meanwhile, a few monomers have been considered to be the necessary precursors for the production of truly absorbable polyesters by ring-opening polymerization. These monomers are glycolide, lactide, and p-dioxanone. Of the condensation-type absorbable polymers, only certain polyanhydrides and polyalkylene oxalates have been recognized as crystalline thermoplastic materials. Most pertinent to the copolymers, subject of this invention, are crystalline absorbable thermoplastic copolymers made by end-grafting low Tg (glass transition temperature) absorbable, so-called "soft" blocks or segments with relatively high Tg crystallizable, chains usually denoted as "hard" blocks or segments. See, for example, U.S. Pat. Nos. 5,554,170; 5,431,679; 5,403,347, 5,236,444; 5,133,739; and 4,429,080. The terms segment and block are used to denote relatively short and long structures of repeat units in the polymeric chain, respectively. In designing the soft segments, or blocks, of the prior art, polar cyclic monomers have been used as precursors to produce these moieties in essentially amorphous, highly flexible form by ring-opening polymerization. However, most of the segments or blocks of the prior art were made to contain small amounts of hydrolytically labile ester linkages derived from glycolide or p-dioxanone to attain a timely absorption of the entire block/segmented copolymer. And in commercial products having the hard-soft segment/block molecular architecture, the hard component of the copolymers have been made primarily or totally of glycolide-derived chains. Unfortunately, having the labile linkage in the soft segments or blocks not only facilitates their absorption, but also causes a premature or early and sudden reduction in the molecular weight of the load-bearing long chains and, hence, an early reduction in breaking strength and related physicomechanical properties of implants based on these copolymers. This provided the incentive to develop the new, linear, semi-crystalline block/segmented copolymers, subject of this invention, wherein the soft blocks or segments are designed to comprise less polar chain sequences formed by step-growth polymerization of acyclic precursors, which are not expected to be absorbable as homopolymers, in order to minimize the hydrolytic instability of the entire block/segmented systems having the more traditional hard components. Accordingly, one aspect of this invention deals with block/segmented copolymers having the soft segment made by step-growth polymerization of an alkane diol and diester which are not known as the common precursors of absorbable homopolymers. In another aspect of this invention, the soft segment/block is made by further end-grafting the aforementioned step-growth alkylene dicarboxylate prepolymer with a cyclic ester and/or carbonate other than those known to provide labile ester linkages such as glycolide and p-dioxane.